A note on Malliavin fractional smoothness for L\'evy processes and approximation

Abstract

Assume a L\'evy process $X$ on the time interval $[0,1]$ that is an $L_2$-martingale and let $Y$ be either its stochastic exponential or $X$ itself. We consider Riemann-approximations of certain stochastic integrals driven by $Y$ and relate the $L_2$-approximation rates to the Malliavin fractional smoothness of the integral to be approximated. The Malliavin fractional smoothness is described by Besov spaces generated with the real interpolation method.